1. Field of the Invention
The invention relates to the field of spherical structures. More particularly, this invention relates to spherical structures assembled from a plurality of convex-concave elements.
2. Description of the Prior Art
Spherical structures as referred to herein include structures that have either a continuously curved or a faceted spherical shape, as well as structures that are semispherical, such as domes, or completely spherical, such as globes. Spherical structures have many and varied applications, but offer particular advantages as spatial enclosures. Not only is the sphere aesthetically pleasing, but it possesses certain structural advantages that make it stronger, more stable, and better able to resist certain forces, such as those resulting from wind, earthquakes, and other natural phenomena, than rectangular structures of comparable size. Nevertheless, despite the structural advantages of spherical over rectangular structures, spherical structures are not commonly used as spatial enclosures and have been constructed primarily for very special purposes. Examples of such special purpose spherical structures are the ancient domes that crown great cathedrals and the arches that impart strength to load-bearing structures such as aqueducts and bridges. Typical housing structures are, however, generally rectangular or cylindrical structures. Several reasons for the lack of use of spherical structures as housing or shelter are based on the fact that such structures are geometrically very complicated and difficult to build; they require special knowledge of spherical geometry and considerable mathematical ability. Thus, making such structures requires specialists and extensive working or shaping of the individual elements, resulting in a structure that is more costly to construct, relative to a rectangular structure of comparable size.
Richard Buckminster Fuller radically altered the task of designing and constructing a spherical structure with his innovative geodesic dome. See U.S. Pat. Nos. 2,682,235 (Fuller; issued 1954), 2,905,113 (Fuller; issued 1959), and 2,914,074 (Fuller; issued 1957). The Fuller geodesic dome is a spherical structure based on a system of regularly-spaced and intersecting gridlines formed by great circles or arcs of a common sphere. The intersecting gridlines form what has come to be known as “geodesic” patterns of nearly equilateral triangles, diamonds, hexagons, or pentagons. The concept of a geodesic dome was a breakthrough in structural form a and design of spherical structures in that it provided a way to construct an approximately spherical structure entirely of planar elements arranged in a straight-edged angular framework, or of flexible elements that were suspended from such an angular framework and allowed to curve to form the overall spherical shape. Examples of such structures include the United States pavilion at the World Expo in Montreal in 1967 and, more recently, sports stadiums like the Astrodome, or the Epcot Center near Disney World in Florida.
The geodesic domes disclosed in U.S. Pat. Nos. 2,682,235 and 2,914,074 comprise a framework of struts and triangular panels arranged within the framework. The structure of a geodesic dome is typically based on a spherical icosahedron, i.e. a polyhedron having twelve vertexes and twenty triangular faces that are superimposed onto a sphere. Each of the twenty faces is a near-equilateral triangle, referred to herein as a “basic triangle”. This geodesic principle of forming a spherical structure from triangular elements only is often referred to by R. B. Fuller as omni-triangulation. The basic triangles of the icosahedron can be further sub-divided along great circle gridlines to create a geodesic structure that approaches more closely a spherical shape. In other words, each basic triangle of the icosahedron can be sub-divided into numerous smaller triangular elements, thereby enabling a relatively large, flat surface of the basic triangle to be broken into many relatively smaller, flat, near-equilateral triangles that can then be assembled to provide the basic triangle with a contour that approaches a curve. The number of sub-divisions of each basic triangle along great circle gridlines with reference to geodesic domes is referred to as the frequency of the dome. Thus, a spherical structure consisting of twenty basic triangles (faces) and twelve vertices has a frequency of one and if the faces are sub-divided by great-circle gridlines that crisscross the triangle in an even grid, the structure has a frequency equal to the number of segments along every side of the basic triangle. For example, if each side of the basic triangle is divided into four segments, the structure has a frequency of four. The higher the frequency of a geodesic structure, the smaller the individual elements become relative to elements in a sphere of the same size having a lower frequency, and the more closely the complete structure can be constructed to approach a spherical form.
Increasing the frequency of the basic structure increases the number and reduces the size of the interdependent triangular elements that make up the basic triangle of the icosahedron. This makes it easier to transport the elements, which can otherwise be difficult if the basic triangle is very large, as is the case with structures such as sports stadiums. One of the disadvantages of the conventional geodesic dome structure is the necessity of constructing a framework of struts in a triangular pattern and then fitting the framework with a skin, or assembling planar elements onto the frame. While the theory behind the geodesic dome appears strikingly simple, enormously complex mathematical operations are required to calculate the precise geometry of the struts and panels. The tolerances required to make a large geodesic dome actually approach those typical of the aircraft industry. This is because six struts must meet precisely at a vertex. Just a few minute errors in the calculation or manufacture of the strut lengths will result in vast discrepancies elsewhere in the structure. Furthermore, although all the triangles in a geodesic grid appear to be of uniform size, the triangles actually differ slightly in size and must be assembled in proper order. This requires that the struts forming the triangles must be identified and assembled in a precise order and, if planar elements are inserted in the framework, each element must be identified and assembled in precise order. Because of the complex calculations required when designing the structural elements of a conventional geodesic dome, it is extremely difficult for persons having ordinary building skills, tools, and materials to construct a dome that has the structural integrity necessary for creating a sturdy and stable structure for shelter.
U.S. Pat. No. 4,270,320 (Chamberlain; issued 1981) discloses a frameless dome structure. This structure comprises circular, spherically curved structural elements, each element having a curvature equal to the curvature of the complete spherical enclosure. The elements are overlapped and attached to each other to create a substantially round, i.e. continuously curved, spherical structure. A key feature of this structure is that each structural element has a spherically curved exterior surface. This requires that the elements be manufactured in precise spherical shapes, that is, the elements must be molded or pressed to form a curved contour that corresponds to the curvature of the complete spherical structure.
A modular dome structure constructed of identical ring-shaped elements that are arranged in even horizontal and vertical rows is disclosed in U.S. Pat. No. 3,959,937 (Spunt; issued 1976). Each ring has four reinforcing ribs to impart rigidity and strength to prevent the rings from deforming into oval shapes. This structure is not a geodesic structure as it is not an omni-triangulated structure. Consequently, the Spunt structure does not offer the structural advantages of strength and flexibility for which the geodesic structure is known.
It may also be desirable to create a structure that is not spherical in shape, in the sense of being a globe, but that is compoundly curved, such as is a pear or a canoe, to form an irregularly curved structure in which the radius of curvature of the structure varies across the outer surface. Compoundly curved structures having a surface with changing radius of curvature are typically based on a plurality of elements having varying curvatures are known, as are compoundly curved structures that are based on hexagonal elements intermixed with a pentagonal element at the vertex. The disadvantages of such structures are similar to those of the conventional geodesic structure—they require very complex mathematical operations.
Spherical structures have well-known uses other than for housing. One such use is that of a globe, i.e., a spherical structure onto which a map of a spherical body is projected. A globe of the earth, for example, displays a map of the earth with the least amount of distortion, because the shape of the globe very closely approximates that of the earth itself. It is not always practical to use a globe, however, and thus, flat two-dimensional maps are often used to show a map of the earth. A two-dimensional map, however, has an inherent disadvantage in that it gives a distorted illustration of a spherical shape. Different types of maps have been devised over the centuries in an attempt to minimize the distortion. An example of one such attempt is the “orange segment” (homolosine) map, which shows the earth laid out as segments of a circle on a two-dimensional plane, with an “empty” space between the upper and lower ends of the segments. This projection of the earth presents less distortion than the map, but some features of interest, such as the areas in or near the polar regions appear disjointed and distorted.
Buckminster Fuller attempted to overcome this problem of map-making by creating the “dymaxion” map. The underlying “globe” on which this map is displayed is not a continuously-curved sphere, but, rather, the spherical icosahedron already known from Fuller's basic geodesic dome structure. Because the icosahedral sphere approximates the shape of the sphere, the map of a spherical body projected onto such a sphere has only a minimum amount of distortion due to the inherent differences in shape between the body to be mapped and the icosahedral sphere. When making the dymaxion map, the icosahedron is so arranged that vertex-to-vertex cuts in the icosahedron do not cross continental landmasses, but are placed, instead, across oceans where the distortion is not as critical. The icosahedron can then be cut along edges of several triangles and laid out flat. The outer contour of the map appears very irregular, but the features of interest on the map, such as the large land masses, are not separated when the icosahedron is cut into a flat map, and the distortion is low. See “The Dymaxion Map”, The Buckminster Fuller Institute. The basic icosahedron of 12 vertexes and 20 faces, however, allows no flexibility in the basic size of the triangles relative to each other and, therefore, is less useful when presenting a map of the heavenly constellations or even of the earth, in which a certain geographic constellation is to be emphasized.
Thomas Smith, Jr. expanded on Fuller's dymaxion map and provided a polyhedral approximation of a spherical body from which a planar map with minimum distortion can be made. See U.S. Pat. No. 5,222,896 (Smith, Jr.; issued 1993). Smith shows a map of the stellar constellations and another one of the earth's moon (FIGS. 10 and 11). The first of these maps comprises a plurality of trapezoidal, pentagonal and hexagonal planar elements that are linked with each other to approximate a spherical structure. The second map comprises a plurality of triangular, F trapezoidal, and pentagonal elements. The elements in both maps vary widely in size and are irregular in shape, i.e., each leg of an element may be a different length. The shapes and sizes of the elements are selected to provide the most sensible and useful presentation of the constellations. The disadvantages of these maps are that, because of the extreme variation in sizes of the elements, the distortion on the map is uneven and may be quite significant.
What is needed, therefore, is a spherical or near-spherical structure constructed of simple structural elements that are easily and inexpensively manufactured. What is further needed is such a structure that can be easily assembled without requiring that the elements be placed or fastened along predetermined great-circle gridlines. What is yet further needed is such a structure for which the necessary materials for a structure of a particular dimension can be easily calculated, without requiring complex mathematical calculations. What is yet further needed is such a structure that is readily adaptable to any compoundly curved shape and provides great versatility for use in many different types of applications.